Selecting Different Airlines for planning a trip to Singapore and Bali

Yes, nothing keeps you from booking every part of your trip separately.

There's a few things to keep in mind though:

• If you travel on separate tickets, your baggage will probably not be checked through. That means you will have to collect your luggage and check it in for the new flight in Singapore. So if you have check-in luggage, you will need to factor in more time for transit. This might especially be an issue if you need a visa to enter Singapore. (Which is the case if you are an Indian national)
• Your connection will not be assured. That means if your first flight is delayed and you miss your connection, you will have to pay again for a new onward flight. If you book everything on one ticket, this will usually be the airlines responsibility.
• I have had airlines question me before when buying a one-way ticket to Singapore. They will probably want to see your onward ticket as proof that you have a way of getting out of Singapore again, so be sure to have that ready.

Note the above is only true if you book the legs separately. If you book them together and one of the flights happens to be operated by another airline with a codeshare agreement, then everything should be fine.